Computing the spectrum of black hole radiation in the presence of high frequency dispersion: an analytical approach

TL;DR

This paper develops an analytical method combining Laplace and WKB techniques to compute the spectrum of black hole radiation with high frequency dispersion, showing the spectrum remains thermal at leading order.

Contribution

It introduces a novel analytical approach to evaluate black hole radiation spectra considering high frequency dispersion effects.

Findings

Spectrum remains thermal at Hawking temperature with high frequency dispersion.

Method effectively handles superluminal and subluminal dispersion modifications.

Out-state is a thermal state at Hawking temperature to leading order.

Abstract

We present a method for computing the spectrum of black hole radiation of a scalar field satisfying a wave equation with high frequency dispersion. The method involves a combination of Laplace transform and WKB techniques for finding approximate solutions to ordinary differential equations. The modified wave equation is obtained by adding a higher order derivative term suppressed by powers of a fundamental momentum scale $k_0$ to the ordinary wave equation. Depending on the sign of this new term, high frequency modes propagate either superluminally or subluminally. We show that the resulting spectrum of created particles is thermal at the Hawking temperature, and further that the out-state is a thermal state at the Hawking temperature, to leading order in $k_0$, for either modification.